(display "ex-1.11\n")

(define (f_rec n)
  (cond ((< n 3) n)
        (else (+ (f_rec (- n 1)) (* 2 (f_rec (- n 2))) (* 3 (f_rec (- n 3)))))))

(display "(f_rec 10) = ")
(display (f_rec 10)) (newline)

(define (f_iter_step n a b c count)
  (cond ((= n count) c)
        (else (f_iter_step n b c (+ (* 3 a) (* 2 b) c) (+ count 1)))))

(define (f_iter n)
  (cond ((< n 3) n)
        (else (f_iter_step n 0 1 2 2))))

(display "(f_iter 10) = ")
(display (f_iter 10)) (newline)


(newline) (display "ex-1.12") (newline)
(define (pascal row col)
  (cond ((= row 1) 1)
        ((= col 1) 1)
        ((= col row) 1)
        (else (+ (pascal (- row 1) (- col 1)) (pascal (- row 1) col)))))

(display "(pascal 6 2) = ")
(display (pascal 6 3)) (newline)


(newline) (display "ex-1.13") (newline)
(display "I was not able to prove this.\n")


(newline) (display "ex-1.14") (newline)
(display "I did that on paper. See page 92 Bullet Journal 2018.")
(newline)

(newline) (display "ex-1.15") (newline)

(define (cube x) (* x x x))
(define (p x) (- (* 3 x) (* 4 (cube x))))
(define (sine angle)
   (if (not (> (abs angle) 0.1))
       angle
       (p (sine (/ angle 3.0)))))

; a) (/ 12.5 3) -> 4.16 -> 1.38888 -> 0.462 -> 0.154 -> 0.051 (count = 5)
; 12.5 / 3^n < 0.1 <=> 125 < 3^n <=> 4.39 < n
(define (count-sine-calls value count)
   (if (> (abs value) 0.1) (count-sine-calls (/ value 3.0) (+ 1 count)) count))
(display "a) Calls for 12.5 = ") (display (count-sine-calls 12.5 0)) (newline)
(display (sine (* 3.14 0.5))) (newline)

; b.  What is the order of growth in space and number of steps
; (as a function of a) used by the process generated by the sine procedure when (sine a) is evaluated?
; O(log_3 a)

(newline) (display "ex-1.16") (newline)

(define (expt-rec x n)
  (cond ((= n 0) 1)
        (else (* x (expt-rec x (- n 1))))))

(define (expt-iter b counter product)
    (cond ((= counter 0) product)
          (else (expt-iter b (- counter 1) (* product b)))))

(define (expt b n) (expt-iter b n 1))

(define (fast-expt b n)
    (cond ((= n 0) 1)
          ((even? n) (square (fast-expt b (/ n 2))))
          (else (* b (fast-expt b (- n 1))))))

(define (even? n)
  (= (remainder n 2) 0))

(define (expt-fast b n)
  (expt-iter-fast b n 1))

(define (expt-iter-fast b n a)
  (cond ((= n 1) (* b a))
        ((even? n) (expt-iter-fast (* b b) (/ n 2) a))
        (else (expt-iter-fast b (- n 1) (* a b)))))

(display "expt-fast 2 37 = ")
(display (expt-fast 2 37))
(newline)


(newline) (display "ex-1.17") (newline)
(define (mul a b)
  (if (= b 0)
      0
      (+ a (mul a (- b 1)))))

(define (double x) (+ x x))
(define (half x) (/ x 2))

(define (mul-rec a b)
  (cond ((= b 0) 0)
        ((even? b) (double (mul-rec a (half b))))
        (else (+ a (mul-rec a (- b 1))))))

(display "mul-rec 17 53 = ")
(display (mul-rec 17 53)) (newline)


(newline) (display "ex-1.18") (newline)
(define (mul-fast-iter b n a)
  (cond ((= n 0) a)
        ((even? n) (mul-fast-iter (double b) (half n) a))
        (else (mul-fast-iter b (- n 1) (+ a b)))))

(define (mul-fast a b) (mul-fast-iter a b 0))

(display "mul-fast 17 53 = ")
(display (mul-fast 17 53))
(newline)


(newline) (display "ex-1.19") (newline)
(define (fib n)
  (fib-iter 1 0 0 1 n))
(define (fib-iter a b p q count)
  (cond ((= count 0) b)
        ((even? count)
         (fib-iter a
                   b
                   (+ (* q q) (* p p))      ; compute p'
                   (+ (* q q) (* 2 p q))    ; compute q'
                   (/ count 2)))
        (else (fib-iter (+ (* b q) (* a q) (* a p))
                        (+ (* b p) (* a q))
                        p
                        q
                        (- count 1)))))

(display "fib 19 = ")
(display (fib 19))
(newline)


(newline) (display "ex-1.20") (newline)
(define (gcd-naiv a b)
  (cond ((= a b) a)
        ((> a b) (gcd-naiv (- a b) b))
        (else (gcd-naiv a (- b a)))))

(display "gcd-naiv 60 14 = ")
(display (gcd-naiv 60 14))
(newline)

(define (gcd a b)
  (if (= b 0)
      a
      (gcd b (remainder a b))))

(display "gcd 60 14 = ")
(display (gcd 60 14))
(newline)
; normaler order
; gcd 206 40
; gcd 40 (r 206 40)
; zero? (r 206 40)
; zero? 6
; gcd (r 206 40) (r 40 (r 206 40))
; zero? (r 40 (r 206 40))
; gcd (r 40 (r 206 40)) (r (r 206 40) (r 40 (r 206 40))))
; etc.

; applicative order
; gcd 206 40
; gcd 40 (r 206 40)
; gcd 40 6
; gcd 6 (r 40 6)
; gcd 6 4
; gcd 4 (r 6 4)
; gcd 4 2
; gcd 2 (r 4 2)
; gcd 2 0 -> 2